ThispaperintroducestheconceptoflocalC-cosins-familyandinverstigateitsbasicproperties,Inparticular,acharacterizationofthecompleteinfinitesimalgeneratorofalocalC-cosinefamilyisobtained,AsanapplieationofthistheorytothesecondordersbstractCauchyproblems,acharacterizationofthelocalC-well-posednessoftheseproblemsisgivenintermsofthelocalC-cosinefamilytheory.
